EXAM 1 - MARCH 4, 2011
Name:
(1) (10 points) Consider the transformation T : R3 → R3 such that T (1, 0, 0) =
(2, 1, 4), T (0, 1, 0) = (4, 3, 6), T (0, 0, 1) = (0, −1, 2).
(a) Determine the null space of T .
(b) If A is the plane formed by span({(2, 5, −3), (−1, −1, 1)}), write T (A)
in parametric form.
1
2
EXAM 1 - MARCH 4, 2011
(2) (10 points) Let
F (t) =
�
(sin t, − cos t)
(sin t, cos t + 2)
t ∈ [0, π]
t ∈ (π, 2π]
(a) Find F � (π), if it is well defined.
(b) Find F �� (π), if it is well defined.
(c) Determine κ(t) everywhere it is defined.
EXAM 1 - MARCH 4, 2011
3
(3) (10 points) Let f (x, y, z) = x2 + y 2 + z 2 . Prove f is differentiable at (1, 1, 1)
with linear transformation T (x, y, z) = 2x + 2y + 2z.
4
EXAM 1 - MARCH 4, 2011
(4) (15 points) Consider the set L(R3 , R2 ) of all linear maps L from R3 to R2
and define addition of L, K ∈ L(R3 , R2 ) the following way:
(v ∈ R3 )
(L + K)(v) = L(v) + K (v)
Define multiplication by a constant c as:
(cL)(v) = c(L(v))
(v ∈ R3 )
(a) Are the linear maps L(x, y, z) = (x, 0), K(x, y, z) = (y, 0), N (x, y, z) =
(x, y) linearly independent? Prove it either way.
(b) Find a basis for L(R3 , R2 ).
(c) What is the dimension of L(R3 , R2 )?
EXAM 1 - MARCH 4, 2011
5
(5) (15 points) Consider the function f : R2 → R that satisfies the following
conditions:
(a) For all fixed x0 ∈ R the function fx0 = f (x0 , y) : R → R is continuous
and;
(b) For all fixed y0 ∈ R the function f y0 = f (x, y0 ) : R → R is continuous
and;
(c) For all fixed x0 ∈ R the function fx0 is monotonically increasing in y,
i.e. if y > y � then, f (x0 , y) > f (x0 , y � ).
Prove f is continuous.
MIT OpenCourseWare
http://ocw.mit.edu
18.024 Multivariable Calculus with Theory
Spring 2011
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.